Plasma-based acceleration is being considered as the basis for building a future linear collider. Nonlinear plasma wakefields have ideal properties for accelerating and focusing electron beams. Preservation of the emittance of nano-Coulomb beams with nanometer scale matched spot sizes in these wakefields remains a critical issue due to ion motion caused by their large space charge forces. We use fully resolved quasistatic particle-in-cell simulations of electron beams in hydrogen and lithium plasmas, including when the accelerated beam has different emittances in the two transverse planes. The projected emittance initially grows and rapidly saturates with a maximum emittance growth of less than 80% in hydrogen and 20% in lithium. The use of overfocused beams is found to dramatically reduce the emittance growth. In conclusion, the underlying physics that leads to the lower than expected emittance growth is elucidated.